Question: Given $ \overrightarrow{PQ}\perp\overrightarrow{PS}$, $ m \angle RPS = 2x + 34$, and $ m \angle QPR = 4x - 28$, find $m\angle QPR$. $P$ $Q$ $S$ $R$
From the diagram, we see that together ${\angle QPR}$ and ${\angle RPS}$ form ${\angle QPS}$ , so $ {m\angle QPR} + {m\angle RPS} = {m\angle QPS}$ Since we are given that $\overrightarrow{PQ}\perp\overrightarrow{PS}$ , we know ${m\angle QPS = 90}$ Substitute in the expressions that were given for each measure: $ {4x - 28} + {2x + 34} = {90}$ Combine like terms: $ 6x + 6 = 90$ Subtract $6$ from both sides: $ 6x = 84$ Divide both sides by $6$ to find $x$ $ x = 14$ Substitute $14$ for $x$ in the expression that was given for $m\angle QPR$ $ m\angle QPR = 4({14}) - 28$ Simplify: $ {m\angle QPR = 56 - 28}$ So ${m\angle QPR = 28}$.